Publication Cover
Applicable Analysis
An International Journal
Volume 94, 2015 - Issue 7
265
Views
2
CrossRef citations to date
0
Altmetric
Articles

The global weak solutions to the Cauchy problem of the generalized Novikov equation

, &
Pages 1334-1354 | Received 05 Apr 2014, Accepted 17 May 2014, Published online: 18 Aug 2014

References

  • Fokas A, Fuchssteiner B. Symplectic structures, their Bäklund transformations and hereditary symmetries. Physica D. 1981;4:47–66.
  • Camassa R, Holm DD. An integrable shallow water equation with peaked solitons. Phys. Rev. Lett. 1993;71:1661–1664.
  • Constantin A, Lannes D. The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations. Arch. Ration. Mech. Anal. 2009;192:165–186.
  • Boutet de Monvel A, Kostenko A, Shepelsky D, Teschl G. Long-time asymptotics for the Camassa-Holm equation. SIAM J. Math. Anal. 2009;41:1559–1588.
  • Constantin A. On the scattering problem for the Camassa-Holm equation. R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 2001;457:953–970.
  • Constantin A, Gerdjikov VS, Ivanov RI. Inverse scattering transform for the Camassa-Holm equation. Inv. Problems. 2006;22:2197–2207.
  • Constantin A, McKean HP. A shallow water equation on the circle. Comm. Pure Appl. Math. 1999;52:949–982.
  • Constantin A. The trajectories of particles in Stokes waves. Invent. Math. 2006;166:523–535.
  • Constantin A, Escher J. Particle trajectories in solitary water waves. Bull. Amer. Math. Soc. 2007;44:423–431.
  • Constantin A, Escher J. Analyticity of periodic traveling free surface water waves with vorticity. Ann. Math. 2011;173:559–568.
  • Toland JF. Stokes waves. Topol. Methods Nonlinear Anal. 1996;7:1–48.
  • Constantin A, Escher J. Wave breaking for nonlinear nonlocal shallow water equations. Acta Math. 1998;181:229–243.
  • Constantin A, Escher J. Well-posedness, global existence and blow-up phenomena for a periodic quasi-linear hyperbolic equation. Comm. Pure Appl. Math. 1998;51:475–504.
  • Danchin R. A note on well-posedness for Camassa-Holm equation. J. Diff. Eqns. 2003;192:429–444.
  • Li YA, Olver PJ. Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation. J. Diff. Eqns. 2000;162:27–63.
  • Yin Z. On the Cauchy problem for an integrable equation with peakon solutions. Illinois J. Math. 2003;47:649–666.
  • Constantin A. Existence of permanent and breaking waves for a shallow water equation: a geometric approach. Ann. Inst. Fourier (Grenoble). 2000;50:321–362.
  • Constantin A, Escher J. Global weak solutions for a shallow water equation. Indiana Univ. Math. J. 1998;47:1527–1545.
  • Constantin A, Molinet L. Global weak solutions for a shallow water equation. Commun. Math. Phys. 2000;211:45–61.
  • Xin Z, Zhang P. On the weak solution to a shallow water equation. Comm. Pure Appl. Math. 2000;53:1411–1433.
  • Novikov VS. Generalizations of the Camassa-Holm equation. J. Phys. A. 2009;42:342002, 14pp.
  • Geng X, Xue B. An extension of integrable peakon equations with cubic nonlinearity. Nonlinearity. 2009;22:1847–1856.
  • Hone ANW, Wang JP. Integrable peakon equations with cubic nonlinearity. J. Phys. A: Math. Theor. 2008;41:372002–372010.
  • Hone ANW, Lundmark H, Szmigielski J. Explicit multipeakon solutions of Novikov’s cubically nonlinear integrable Camassa-Holm equation. Dyn. Partial Diff. Eqns. 2009;6:253–289.
  • Grayshan K. Peakon solutions of the Novikov equation and properties of the data-to-solution map. J. Math. Anal. Appl. 2013;397:515–521.
  • Himonas AA, Holliman C. The Cauchy problem for the Novikov equation. Nonlinearity. 2012;25:449–479.
  • Ni L, Zhou Y. Well-posedness and persistence properties for the Novikov equation. J. Diff. Eqns. 2011;250:3002–3201.
  • Yan W, Li Y, Zhang Y. The Cauchy problem for the integrable Novikov equation. J. Diff. Eqns. 2012;253:298–318.
  • Lai SY, Li N, Wu YH. The existence of global strong and weak solutions for the Novkov equation. J. Math. Anal. Appl. 2013;399:682–691.
  • Wu X, Yin Z. Global weak solutions for the Novikov equation. J. Phys. A: Math. Theor. 2011;44:055202, 17pp.
  • Malek J, Necas J, Rokyta M, Ruzicka M. Weak and measure-valued solutions to evolutionary PDEs. London: Chapman & Hall; 1996.
  • Kato T, Ponce G. Commutator estimates and the Euler and Navier-Stokes equations. Comm. Pure Appl. Math. 1988;41:891–907.
  • Chemin JY. Phase space analysis of partial differential equations, Proceedings. CRM series. Pisa: Centro Edizioni, Scunla Normale Superiore; 2004. p. 53–136.
  • Zhao Y, Li Y, Yan W. Local Well-posedness and persistence property for the generalized Novikov equation. Discr. Contin. Dyn. Syst. Ser. A. 2014;34:803–820.
  • Natanson IP. Theory of functions of a real variable. New York (NY): F. Ungar Publ. Co.; 1964.

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.